# QuanVI: Score-based Variational Inference via Quantum Maximally Mixed States

arXiv: https://arxiv.org/abs/2609.39164 · HTML: https://mlsanity.com/paper/2609.39164/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 33 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 19 of 19)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Automatic Differentiation Variational Inference](https://mlsanity.com/paper/1603.00788/) (2016)
- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning](https://mlsanity.com/paper/1907.03741/) (2019)
- [Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives](https://mlsanity.com/paper/1708.09165/) (2017)
- [Tensor Regression Networks](https://mlsanity.com/paper/1707.08308/) (2017)
- [Stochastic Backpropagation and Approximate Inference in Deep Generative Models](https://mlsanity.com/paper/1401.4082/) (2014)
- [Tensor-Train Density Estimation](https://mlsanity.com/paper/2108.00089/) (2021)
- [QuanTA: Efficient High-Rank Fine-Tuning of LLMs with Quantum-Informed Tensor Adaptation](https://mlsanity.com/paper/2406.00132/) (2024)
- [Batch and match: black-box variational inference with a score-based divergence](https://mlsanity.com/paper/2402.14758/) (2024)
- [Semi-Implicit Variational Inference via Score Matching](https://mlsanity.com/paper/2308.10014/) (2023)
- [Variational Inference with Gaussian Score Matching](https://mlsanity.com/paper/2307.07849/) (2023)
- [EigenVI: score-based variational inference with orthogonal function expansions](https://mlsanity.com/paper/2410.24054/) (2024)
- [Generative Learning of Continuous Data by Tensor Networks](https://mlsanity.com/paper/2310.20498/) (2023)
- [Variational Inference with Mixtures of Isotropic Gaussians](https://mlsanity.com/paper/2506.13613/) (2025)
- [Fisher meets Feynman: score-based variational inference with a product of experts](https://mlsanity.com/paper/2510.21598/) (2025)

## Abstract (arXiv)

Score-based variational inference (VI) provides an alternative to Kullback--Leibler (KL)-based VI by minimizing the Fisher divergence between the variational distribution and the target. A prior score-VI approach formulates this optimization as an eigenvalue problem, with the variational distribution constructed from low-energy eigenstates. However, this eigenvalue-based formulation faces two high-dimensional obstacles: an intractably large parameter count due to exponential scaling and non-uniqueness of individual eigenvectors in degenerate or nearly degenerate low-energy subspaces. We propose QuanVI, a scalable quantum-inspired algorithm that combines a mixed-state density-operator formulation with a quantum tensor network (QTN) parameterization using the matrix product operator (MPO) structure. In degenerate low-energy subspaces, the density-operator formulation represents the subspace by its maximally mixed state rather than relying on a non-unique individual eigenvector, while the QTN parameterization compresses the density operator to avoid exponential parameter growth. Experiments and ablations show that QuanVI agrees with exact solutions in low dimensions and scales to high-dimensional synthetic and Bayesian posterior-approximation benchmarks, including challenging non-Gaussian targets.